Textbook Question
Faster than a calculator Use the approximation (1 + x)ᵏ ≈ 1 + kx to estimate the following.
a. (1.0002)⁵⁰
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 3, Problem 3.5.53a
By computing the first few derivatives and looking for a pattern, find the following derivatives.
a. d⁹⁹⁹/dx⁹⁹⁹ (cos x)
Verified step by step guidance1
Start by computing the first derivative of cos(x). The derivative of cos(x) with respect to x is -sin(x).
Compute the second derivative. The derivative of -sin(x) is -cos(x).
Compute the third derivative. The derivative of -cos(x) is sin(x).
Compute the fourth derivative. The derivative of sin(x) is cos(x).
Notice the pattern: the derivatives cycle every four steps: cos(x), -sin(x), -cos(x), sin(x). Use this pattern to determine the 999th derivative by finding the remainder when 999 is divided by 4.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives of Trigonometric Functions
Understanding the derivatives of basic trigonometric functions is essential. The derivative of cos(x) is -sin(x), and the derivative of sin(x) is cos(x). This cyclical pattern continues, with the derivative of -sin(x) being -cos(x), and the derivative of -cos(x) being sin(x). Recognizing this cycle helps in computing higher-order derivatives.
Recommended video:
6:04
Introduction to Trigonometric Functions
Higher-Order Derivatives
Higher-order derivatives involve taking the derivative of a function multiple times. For trigonometric functions like cos(x), the derivatives repeat in a cycle every four derivatives. This means that the nth derivative can be determined by finding the remainder of n divided by 4, which indicates the position in the cycle.
Recommended video:
02:42Higher Order Derivatives
Pattern Recognition in Derivatives
Identifying patterns in derivatives is crucial for efficiently computing higher-order derivatives. By observing the cyclical nature of the derivatives of cos(x), one can predict the 999th derivative by recognizing that it corresponds to the third position in the cycle, which is -sin(x). This pattern recognition simplifies the computation process significantly.
Recommended video:
05:44Derivatives
Related Practice
Textbook Question
Hauling in a dinghy A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow. The rope is hauled in at the rate of 2 ft/sec.
a. How fast is the boat approaching the dock when 10 ft of rope are out?
Textbook Question
Consider the function f graphed here. The domain of f is the interval [−4, 6] and its graph is made of line segments joined end to end.
" style="" width="350">
b. Graph the derivative of f. The graph should show a step function.
Textbook Question
In Exercises 47 and 48, find an equation for
(b) the horizontal tangent line to the curve at Q.
" style="" width="200">
Textbook Question
The accompanying figure shows the velocity v = ds/dt = f(t) (m/sec) of a body moving along a coordinate line.
a. When does the body reverse direction?
Textbook Question
The folium of Descartes (See Figure 3.27)
b. At what point other than the origin does the folium have a horizontal tangent line?